Unsteady Stagnation Point Flow and Heat Transfer over a

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2012, Article ID 781845, 12 pagesdoi:10.1155/2012/781845

Research ArticleUnsteady Stagnation Point Flow and HeatTransfer over a Stretching/Shrinking Sheetwith Suction or Injection

M. Suali,1 N. M. A. Nik Long,1, 2 and N. M. Ariffin1, 2

1 Department of Mathematics, Universiti Putra Malaysia, Selangor, 43400 Serdang, Malaysia2 Institute for Mathematical Research, Universiti Putra Malaysia, Selangor, 43400 Serdang, Malaysia

Correspondence should be addressed to M. Suali, [email protected]

Received 29 February 2012; Accepted 12 April 2012

Academic Editor: Srinivasan Natesan

Copyright q 2012 M. Suali et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The unsteady stagnation point flow and heat transfer over a stretching/shrinking sheet withsuction/injection is studied. The governing partial differential equations are converted intononlinear ordinary differential equations using a similarity transformation and solved numerically.Both stretching and shrinking cases are considered. Results for the skin friction coefficient, localNusselt number, velocity, and temperature profiles are presented for different values of thegoverning parameters. It is found that the dual solutions exist for the shrinking case, whereas thesolution is unique for the stretching case. Numerical results show that the range of dual solutionsincreases with mass suction and decreases with mass injection.

1. Introduction

The study of flow and heat transfer over a stretching/shrinking sheet receives considerableattention from many researchers due to its variety of application in industries such asextrusion of a polymer in a melt spinning process, manufacturing plastic films, wire drawing,hot rolling, and glass fiber production. Sakiadis [1, 2] reported the flow field analysis wherethe stretched surface was assumed to move with uniform velocity, and similarity solutionswere obtained for the governing equations. Crane [3] gave a closed-form solution for steadytwo-dimensional flow of an incompressible viscous fluid caused by the stretching of an elasticsheet, which moves in its own plane with a velocity which varies linearly with distancefrom a fixed point. Chiam [4] investigated the steady two-dimensional stagnation pointflow towards a stretching surface in the case when the stretching velocity is identical tothe free stream velocity. P.S. Gupta and A.S. Gupta [5] extended the work of Crane [3] by

2 Journal of Applied Mathematics

investigating the effect of mass transfer on a stretching sheet with suction or blowing forlinear surface velocity subject to uniform temperature. Mahapatra and Gupta [6] investigatedthe steady two-dimensional magnetohydrodynamics (MHDs) stagnation point flow of anincompressible viscous electrically conducting fluid toward a stretching surface, the flowbeing permeated by a uniform transverse magnetic field. In studying similar problem toChiam [4], Mahapatra and Gupta [7] observed that the structure of the boundary layerdepends on the stretching sheet parameters and the angle of incidence. Mahapatra and Gupta[8] obtained the exact similarity solution of the Navier-Stokes equations by consideringthe combination of both stagnation flow and stretching surface and observed that the flowdisplays a boundary-layer structure when the stretching velocity of the surface is lessthan the free stream velocity. Lok et al. [9] extended the Mahapatra and Gupta’s work[8] to the oblique stagnation flow and found that the free stream obliqueness is the shiftof the stagnation point toward the incoming flow and depends on the inclination angle.Miklavcic and Wang [10] obtained the solution for steady two-dimensional, as well asaxisymmetric viscous, flow over a shrinking sheet. Fang and Zhang [11] obtained the closed-form solution for steady MHD flow over a shrinking surface subject to applied suction. Wang[12] investigated the steady two-dimensional and axisymmetric stagnation point flow withheat transfer over a shrinking sheet and found that solutions do not exist for the largershrinking rates. Ishak et al. [13] extended the work of Mahapatra and Gupta [8] to the obliquestagnation flow and a shrinking sheet in a micropolar fluid. It was found in these problemsthat the solutions for a shrinking sheet are non-unique. Bachok et al. [14] considered thesimilarity solutions of stagnation point flow and heat transfer over a warm, laminar liquidflow to a melting stretching or shrinking sheet. Recently, Patel and Timol [15] analyzed thesteady two-dimensional stagnation point flow and heat transfer of a non-Newtonian fluid. Inthe unsteady flow problem, Wang [16] studied the concept of the flow and investigated thebehavior of liquid film on an unsteady stretching surface. Andersson et al. [17] presenteda new similarity solution for the temperature field on an unsteady stretching surface bytransforming the time-dependent thermal energy equation into an ordinary differentialequation. Elbashbeshy and Bazid [18] presented an exact similarity solution for unsteadyflow and heat transfer over a horizontal stretching surface and studied the effect of variousgoverning parameters such as the Prandtl number and unsteadiness parameter whichdetermine the velocity and temperature profiles and heat transfer coefficient. Later, Fang[19] extended the flow over a shrinking sheet to power law surface velocity and obtainedthe multiple solutions for certain mass transfer with controlling parameters. Tsai et al. [20]studied numerically the nonuniform heat source or sink effect on the flow and heat transferfrom an unsteady stretching sheet through a quiescent fluid. The results showed that theheat transfer rate and the skin friction increase as the unsteadiness parameter increases, but,as the space- and temperature-dependent parameters for heat source/sink increase, the heattransfer rate and the skin friction decrease. Sajid and Hayat [21] investigated the effect ofthe MHD for two-dimensional and axisymmetric shrinking sheet. Later, Fang et al. [22]found that multiple solutions exist for a certain range of mass suction and unsteadinessparameters for the unsteady viscous flow over a continuously shrinking sheet. Recently,Adhikary and Misra [23] investigated the unsteady two-dimensional hydromagnetic flowand heat transfer of an incompressible viscous fluid. Ali et al. [24] presented the influence ofrotation, unsteadiness, and mass suction parameters on the reduced skin friction coefficientsas well as the lateral velocity and velocity profiles for the unsteady viscous flow over ashrinking sheet with mass transfer in a rotating fluid. Recently, Fang et al. [25] investigatedthe boundary layers of an unsteady incompressible stagnation point flow with mass transfer

Journal of Applied Mathematics 3

whereas Bhattacharyya [26] found that the dual solutions for velocity distribution exist forcertain value of velocity ratio parameter for the dual solutions in unsteady stagnation pointflow over a shrinking sheet. Very recently, Nik Long et al. [27] found that multiple solutionsexist for a certain range of ratio of the shrinking velocity to the free stream velocity whichagain depends on the unsteadiness parameter for the unsteady stagnation point flow andheat transfer over a stretching/shrinking sheet. The aim of this work is to extend the paperby Nik Long et al. [27] to the case of unsteady stagnation point flow of an incompressibleviscous fluid by considering both stretching and shrinking sheets with suction or injection.The momentum and heat equation are solved numerically, and the characteristics of the floware obtained.

2. Mathematical Formulation

Consider the unsteady stagnation point flow over a stretching or shrinking sheet immersedin an incompressible viscous fluid of ambient temperature T∞. It is assumed that the freestream velocity is in the form U∞(x, t) = ax(1 − λt)−1, the sheet is stretched with velocityUw(x, t) = bx(1 − λt)−1, and the surface temperature is Tw(x, t) = T∞ + cx(1 − λt)−1. The x-axis runs along the sheet, and the y-axis is measured normal to it. Having these assumptions,along with the boundary-layer approximations and neglecting the viscous dissipation, thegoverning equations are given by

∂u

∂x+∂v

∂y= 0, (2.1)

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y=∂U∞∂t

+U∞∂U∞∂x

+ ν∂2u

∂y2,

∂T

∂t+ u

∂T

∂x+ v

∂T

∂y= α

∂2T

∂y2,

(2.2)

with the boundary conditions

u = Uw, v = Vw, T = Tw, at y = 0,

u −→ U∞, T −→ T∞, as y −→ ∞,(2.3)

where u and v are velocity components in x and y directions; respectively, ν is kinematicviscosity; α is thermal diffusivity; T is fluid temperature. Introducing the following similaritytransformations

ψ(x, y, t

)=(

aν

1 − λt)1/2

xf(η),

η(y, t

)=(

a

ν(1 − λt))1/2

y,

θ(η)=

T − T∞Tw − T∞ ,

(2.4)


where η is similarity variable and ψ is stream function defined as u = ∂ψ/∂y and v = −∂ψ/∂x,thus we have

u =ax

1 − λtf′(η

), v = −

(aν

1 − λt)1/2

f(η); (2.5)

therefore, the mass transfer velocity Vw can take the form

Vw(t) = −(

aν

1 − λt)1/2

f0, (2.6)

where prime denotes differentiation with respect to η. With these values of u and v, (2.1)is identically satisfied whereas (2.2) reduce to the following nonlinear ordinary differentialequations

f ′′′ + ff ′′ + 1 − f ′2 +A(

1 − f ′ − 12ηf ′′

)= 0,

1Prθ′′ + fθ′ − f ′θ −A

(θ +

η

2θ′)= 0.

(2.7)

The boundary conditions (2.3) become

f(0) = f0, f ′(0) =b

a= ε, θ(0) = 1, θ(∞) −→ 0, f ′(∞) −→ 1, (2.8)

where ε(= b/a) is the ratio of stretching/shrinking velocity parameter and free streamvelocity parameter, f0 > 0 and f0 < 0 are the suction and injection parameters, respectively,Pr = v/α is the Prandtl number, and A = λ/a is unsteadiness parameter. The quantitiesof physical interest to be obtained are the effect of suction or injection on the skin frictioncoefficient f ′′(0) and the local Nusselt number −θ′(0) which are defined as

Cf =τw

ρU2∞/2, Nux =

xqwk(Tw − T∞) , (2.9)

where the surface shear stress τw and the surface heat flux qw are defined as

τw = μ(∂u

∂y

)

y=0, qw = −k

(∂T

∂y

)

y=0, (2.10)

with μ and k being the dynamic viscosity and thermal conductivity, respectively. Using thesimilarity variables equation (2.4), we obtain

12CfRe1/2

x = f ′′(0),Nux

Re1/2x

= −θ′(0), (2.11)


f′′ (

0)

1

0

−1

−2

−3

−4

10−1−2−3−4−5−6−7−8

A

f0 = −0.5, −0.1, 0, 0.5

Figure 1: Solution domain for momentum boundary layers under different mass transfer parameters whenε = 0.01.

where Rex = U∞x/ν is local Reynolds number. It should be noticed that (2.7) reduce tothose of Wang [12] and Nik Long et al. [27] when A = 0 (steady-state flow) and A = 0.01,respectively.

3. Results and Discussion

The nonlinear ordinary differential equations (2.7) subject to the boundary conditions (2.8)are solved numerically using the shooting method where we convert the boundary valueproblem into an initial value problem. The numerical results are given to carry out aparametric study showing the influence of the nondimensional parameters: the unsteadinessparameters A, ε, Prandtl number Pr, and f0. For the validation of numerical results, the caseA = 0 and Pr = 0.7 with no effect of suction or injection (f0 = 0) are considered and comparedto those of Wang [12]. The quantitative comparison is shown in Table 1 for the variation ofε with parameter f0 and found to be in favorable agreement. Based on our computations,the critical values of ε, εc with parameter f0 are presented in Table 2 and compare wellwith the previous results reported by Nik Long et al. [27]. It is observed that the valuesof |εc| increase as f0 increases. Hence, suction delays the boundary layer separation, whereasinjection accelerates it.

The solution domains for f ′′(0) are shown in Figures 1 and 2 for different mass transferparameters f0 and ε, respectively. It is seen that there exist dual solutions for Ac < A < 0where Ac is a critical number dependent on f0. The solution is unique for (A > 0)∪ (A = Ac),and there is no solution for (A < Ac). The existence of the dual solution in our case is inagreement with [25, 28, 29]. The effect of ε and the mass suction (f0 > 0) enlarge the solution


0−4 −2−6−12 −10 −8

f′′ (

0)

1

0

−1

−2

−3

−4

A

ε = −0.5, 0.01, 0.5

Figure 2: Solution domain for momentum boundary layers for different values of ε when f0 = 0.

domain, and the mass injection (f0 < 0) reduces it. The velocity gradient decreases withdecreasing values of A for the first and second solutions from positive to negative values off ′′(0), which indicates that there exist multiple regions for the momentum boundary layer.Figures 3 and 4 show the relationship between the skin friction coefficient and the localNusselt number with ε and for the various values of f0. It is observed that there is no solutionsfor ε < εc, the dual solutions exist for εc < ε ≤ −1.0, and the solutions are unique for ε > −1.0,which means that the boundary layer separates from the surface. Figures 5 and 6 show theskin friction coefficient and the local Nusselt number with f0, respectively, for ε = −1.18and A = 0.01, 0.04. For these values of ε, critical value (f0,c) of f0 being negative at which,there is a saddle node bifurcation. The dual solutions exist beyond the critical number whichis f0 > f0,c, but there is no solution for f0 < f0,c. This behavior of solution indicates thatthe solutions exist for a certain interval of ε the case of injection, whereas no such intervalappears for the suction, with both branches of solutions continuing to large value of f0. Theskin friction coefficient f ′′(0) and the local Nusselt number −θ′(0) increase as the parameterA increases and decrease as f0 increases. The samples of velocity and temperature profiles arepresented in Figures 7 and 8, respectively, for different values of f0, ε = −1.18, and A = 0.01.These figures show that the boundary conditions (2.8) for (2.7) are satisfied and approachedinfinity asymptotically, which support the numerical results presented in Figures 3, 4, 5, and6 and exhibit the existence of the dual solution. We observed that the temperature profilesincrease as the values of f0 increase. The same behavior of solution is observed for the velocityprofiles.

4. Conclusion

A numerical study is performed for the problem of unsteady two-dimensional stagnationpoint flow and heat transfer over a stretching and shrinking sheets with suction or injection.


1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

−1 −0.5 0 0.5 1

ε

f′′ (

0)

f0 = −0.1, 0, 0.1

Pr = 0.7

(−1, 0)

Figure 3: Skin friction coefficient f ′′(0) as a function of ε for different values of f0 when A = 0.01.

−1 −0.5 0 0.5 1ε

f0 = −0.1, 0, 0.1

Pr = 0.7

−θ′ (

0)

1

0

−1

−2

−3

−4

−5

−6

−7

−8

Figure 4: Local Nusselt number −θ′(0) as a function of ε for different values of f0 when A = 0.01.


−0.2 0 0.2 0.4 0.6 0.8 1

3

2

1

0

f0

A = 0.01A = 0.04

ε = −1.18

A = 0.01, 0.04

f′′ (

0)

Figure 5: Skin friction coefficient f ′′(0) as a function of f0 for different values of A.

−0.2 0 0.2 0.4 0.6 0.8 1f0

A = 0.01A = 0.04

−θ′ (

0)

8

6

4

2

0

−2

ε = −1.18

Figure 6: Local Nusselt number −θ′(0) as a function of f0 for different values of A when Pr = 0.7.


0 1 2 3 4 5 6 7η

f′ (η)

f0 = −0.1, 0, 0.1

f0 = −0.1, 0, 0.1

1

0.5

0

−0.5

−1

First solutionSecond solution

Figure 7: The velocity profiles f ′(η) for different values of f0 when A = 0.01, ε = −1.18.

First solutionSecond solution

f0 = −0.1, 0, 0.1

0 1 2 3 4 5 6 87

0

1

2

3

4

5

6

8

7

η

θ(η)

Figure 8: The temperature profiles θ(η) for different values of f0 when A = 0.01, ε = −1.18, and Pr = 0.7.


Table 1: Variation of f ′′(0) with ε when f0 = 0.

A εWang [12] Present

First solution Second solution First solution Second solution

0 3 −4.276545

0.2 1.05113 1.051130

0.1 1.14656 1.146561

−0.1 1.30860 1.308602

−0.25 1.40224 1.402240

−0.5 1.49567 1.495672

−1.0 1.32882 0 1.328817 0

−1.15 1.08223 0.116702 1.082232 0.116702

Table 2: The values of εc for different values of A and f0.

A f0 Nik Long et al. [27] Present

−0.3 −0.265432

−2 0 −0.439398

0.3 −0.638668

−0.3 −0.816678

−0.5 0 −0.956566

0.3 −1.139657

0.01 0 −1.2536 −1.253600

−0.3 −1.464110

0.5 0 −1.601111

0.3 −1.725011

−0.3 −1.969922

2 0 −2.138974

0.3 −2.383360

The similarity transformation is used to reduce the partial differential equations intononlinear ordinary differential equations. The effects of the mass suction or mass injectionparameters f0, governing parameters A, and ε on the fluid flow and heat transfer character-istics have been discussed, and the numerical results obtained are comparable well with thepreviously reported cases. It was found that the suction (f0 > 0) widens the range of ε forwhich solution exists, whereas the injection (f0 < 0) acts in the opposite manner. Moreover,for the shrinking sheet, the dual solutions exist whereas for the stretching sheet the solutionis found to be unique for all ε.

Acknowledgment

The second author would like to thank the Ministry of Higher Education (MOHE) for theGrant no. 03-11-08-670FR.


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